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BAGHDAD, Iraq (CNN) -- Vote counting in Iraq began Thursday night after a surprisingly high number of voters turned out to choose the nation's first full-term parliament since Saddam Hussein's ouster.

Results from the 33,000 polling stations probably won't be available for "two weeks or more," said Farid Ayar, spokesman for the Independent Electoral Commission of Iraq.

Officials estimated about 10 million of the 15 million registered voters participated in the elections for a 275-seat parliament called the Council of Representatives.

Among those who streamed to the polls were Sunni Arabs, who had stayed away in previous elections only to find they barely had a voice in government.

The high turnout was remarkable considering curfews, bulked-up security, border closings, road closures and traffic bans across the country.

The turnout was so heavy in some areas that election officials gave provincial governments discretion to keep polls open an hour past the 5 p.m. closing time.

"It's been a good day for Iraq," said Laith Kubba, a top aide to transitional Prime Minister Ibrahim al-Jaafari.

Only scattered violence was reported. An explosion caused two deaths -- an Iraqi soldier and a civilian -- in Baquba in Diyala province.

On election eve in Ramadi, a bomb killed a U.S. Marine assigned to the 2nd Marine Logistics Group, II Marine Expeditionary Force (Forward). The death brought the number of U.S. troops killed in the Iraq war to 2,152.

The U.S. military said two 127 mm rockets were fired in central Baghdad Thursday morning, one of them landing behind a polling station. Three people were wounded.

At four different Sadr City polling stations in eastern Baghdad, three armed terrorists were blamed for harassing voters, election officials said.

Nonetheless, one volunteer poll worker in Baquba deemed it "a special day."

In some cases, voters had to take long walks to get to polls. Many were seen happily thrusting their purple ink-stained fingers at photographers -- the colored fingers a symbol of Iraq's free elections.

In Ramadi, local clerics used mosque sound systems, usually reserved for calls to prayer, to urge people to vote.

A celebratory atmosphere took hold in some locations. In the eastern Ramadi neighborhood of Sufiya, candy was handed out as people came to vote.

Officials pleased

The presence of Sunni Arab voters pleased U.S. and Iraqi officials, who believe their acceptance of the new post-Hussein government will quell the Sunni-dominated insurgency.

U.S. Ambassador to Iraq Zalmay Khalilzad said that successfully integrating the Sunni Arab community into the political process was a factor that would contribute to the start of a pullout of U.S. forces after the elections.

"This is a major step forward in achieving our objective," said President Bush, referring to the U.S. goal of creating a democratic Iraq and an ally.

"This is a crucial part of the war on terror," he added.

Thousands of Iraq expatriates voted in 15 countries -- including the United States -- over the past three days.

More than 19 political coalitions ran in the election, along with 307 political entities -- either independent parties or individual candidates. (Find out how system works)

Expected to fare well are the ruling coalitions during the transitional period -- the Shiite-led United Iraqi Alliance and the Kurdish bloc.

Alleged violations

Iraqi election officials were looking into reports of voter intimidation and campaigning transgressions.

Sources said that at some polling places police told people via megaphone to vote for list 555, which is the ruling United Iraqi Alliance.

The Washington Post reported the large turnout in the the insurgent hotbed of Falluja, in Anbar province, caused polling places to run out of ballots and ballot boxes, and election workers tried to replenish supplies.

Similar problems were reported elsewhere in Anbar province, and the unstable "security situation" in the province prevented some polling stations from opening, said Ayar, the election commission spokesman.

Commission member Safwat Rashid Sidai said he would investigate reports that a number of voting stations failed to open in Yusufiya, southwest of Baghdad.

Director-general of the commission, Ali al-Lami, dismissed reports of truckloads of fake ballots coming from Iran, saying the Hyundai trucks mentioned in news stories belonged to the commission and were carrying voting materials to polling stations.